Proposition 1.

Let XXX be a closure space with ccc the associated closure operator. Define a “closed set” of XXX as a subset AAA of XXX such that Ac=AsuperscriptAcAA^{c}=A, and an “open set” of XXX as the complement of some closed set of XXX. Then the collection ??\mathcal{T} of all open sets of XXX is a topology on XXX.

Proof.

Since ∅c=∅superscriptc\varnothing^{c}=\varnothing, ∅\varnothing is closed. Also, X⊆XcXsuperscriptXcX\subseteq X^{c} and Xc⊆XsuperscriptXcXX^{c}\subseteq Ximply that Xc=XsuperscriptXcXX^{c}=X, or XXX is closed. If A,B⊆XABXA,B\subseteq X are closed, then (A∪B)c=Ac∪Bc=A∪BsuperscriptABcsuperscriptAcsuperscriptBcAB(A\cup B)^{c}=A^{c}\cup B^{c}=A\cup B is closed as well. Finally, suppose AisubscriptAiA_{i} are closed. Let B=⋂AiBsubscriptAiB=\bigcap A_{i}. For each iii, Ai=B∪AisubscriptAiBsubscriptAiA_{i}=B\cup A_{i}, so Ai=Aic=(B∪Ai)c=Bc∪Aic=Bc∪AisubscriptAisuperscriptsubscriptAicsuperscriptBsubscriptAicsuperscriptBcsuperscriptsubscriptAicsuperscriptBcsubscriptAiA_{i}=A_{i}^{c}=(B\cup A_{i})^{c}=B^{c}\cup A_{i}^{c}=B^{c}\cup A_{i}. This meansBc⊆AisuperscriptBcsubscriptAiB^{c}\subseteq A_{i}, or Bc⊆⋂Ai=BsuperscriptBcsubscriptAiBB^{c}\subseteq\bigcap A_{i}=B. But B⊆BcBsuperscriptBcB\subseteq B^{c} by definition, so B=BcBsuperscriptBcB=B^{c}, or that ⋂AisubscriptAi\bigcap A_{i} is closed.
∎

??\mathcal{T} so defined is called the closure topology of XXX with respect to the closure operator ccc.

Remarks.

1.

A closure space can be more generally defined as a set XXX together with an operatorcl:P⁢(X)→P⁢(X)normal-:clnormal-→PXPX\operatorname{cl}:P(X)\to P(X) such that clcl\operatorname{cl}satisfies all of the Kuratowski’s closure axioms where the equal sign “==” is replaced with set inclusion “⊆\subseteq”, and the preservation of ∅\varnothing is no longer assumed.

2.

Even more generally, a closure space can be defined as a set XXX and an operator clcl\operatorname{cl} on P⁢(X)PXP(X) such that

A⊆cl⁡(A)AclAA\subseteq\operatorname{cl}(A),

cl⁡(cl⁡(A))⊆cl⁡(A)clclAclA\operatorname{cl}(\operatorname{cl}(A))\subseteq\operatorname{cl}(A), and

clcl\operatorname{cl} is order-preserving, i.e., if A⊆BABA\subseteq B, then cl⁡(A)⊆cl⁡(B)clAclB\operatorname{cl}(A)\subseteq\operatorname{cl}(B).

It can be easily deduced that cl⁡(A)∪cl⁡(B)⊆cl⁡(A∪B)clAclBclAB\operatorname{cl}(A)\cup\operatorname{cl}(B)\subseteq\operatorname{cl}(A\cup B). In general however, the equality fails. The three axioms above can be shown to be equivalent to a single axiom:

In a closure space XXX, a subset AAA of XXX is said to be closed if cl⁡(A)=AclAA\operatorname{cl}(A)=A. Let C⁢(X)CXC(X) be the set of all closed sets of XXX. It is not hard to see that if C⁢(X)CXC(X) is closed under∪\cup, then clcl\operatorname{cl} “distributes over” ∪\cup, that is, we have the equality cl⁡(A)∪cl⁡(B)=cl⁡(A∪B)clAclBclAB\operatorname{cl}(A)\cup\operatorname{cl}(B)=\operatorname{cl}(A\cup B).

4.

Also, cl⁡(∅)cl\operatorname{cl}(\varnothing) is the smallest closed set in XXX; it is the bottom element in C⁢(X)CXC(X). This means that if there are two disjoint closed sets in XXX, then cl⁡(∅)=∅cl\operatorname{cl}(\varnothing)=\varnothing. This is equivalent to saying that ∅\varnothing is closed whenever there exist A,B⊆XABXA,B\subseteq X such that cl⁡(A)∩cl⁡(B)=∅clAclB\operatorname{cl}(A)\cap\operatorname{cl}(B)=\varnothing.

5.

Since the distributivity of clcl\operatorname{cl} over ∪\cup does not hold in general, and there is no guarantee that cl⁡(∅)=∅cl\operatorname{cl}(\varnothing)=\varnothing, a closure space under these generalized versions is a more general system than a topological space.